The invention relates to a method of deriving a time-averaged moment of a convolution profile from dynamic input and arrival profiles, the arrival profile being related to the input profile according to convolution with the convolution profile.
The article xe2x80x9cHigh resolution measurement of cerebral blood flow using intravascular tracer bolus passages, part I: mathematical approach and statistical analysisxe2x80x9d by L. Østergaard et al in MRM 36 (1996), 715-725, concerns the study of the blood flow through the brain. For such an examination a contrast agent is administered to a patient to be examined; for example, a contrast liquid is injected into a blood vessel. The concentration in which the contrast agent is fed, via an artery, to a part of the body of the patient to be examined, is determined in a time resolved manner. The time-dependent concentration of supplied contrast agent is referred to as the arterial input. Subsequently, the concentration of the contrast agent encountered in veins of the part of the body of the patient to be examined is determined in a time resolved manner. The time-dependent concentration of contrast agent in the artery is called the venous output in said article. A convolutive relation exists between the venous output and the arterial input. The convolution kernel is then the distribution of transit times through the network of blood vessels connecting the artery via which the contrast agent is supplied to veins in which the contrast agent is encountered somewhat later. This distribution of transit times is referred to as the transport function.
A method of the kind disclosed in the heading of Claim 1 is known from the cited article by L. Østergaard et al.; therein, the arterial input and the venous output are examples of the dynamic input profile and the arrival profile, respectively, and the transport function is an example of the convolution profile. According to the known method it is necessary to deconvolute the venous output so as to determine the transport function. The average transit time of the contrast medium through the network of blood vessels and the time-dependent concentration of contrast agent in a selected volume (volume-of-interest) are calculated on the basis of the transport function.
For the known method to yield reliable results it is necessary to determine the arterial input particularly accurately. It has been found nevertheless that, even when the arterial input is extremely accurately measured, the reliability of the calculated results is rather disappointing. Moreover, the known method involves rather complex and time-consuming calculations.
It is an object of the invention to provide a method whereby perfusion quantities can be accurately determined in a simpler manner in comparison with the known method.
This object is achieved by means of a method according to the invention which is characterized in that:
a time-averaged moment of a dynamic input profile and
a time-averaged moment of a dynamic arrival profile are determined, and that
the time-averaged moment of the convolution profile is calculated from the time-averaged moments of the dynamic input profile and the time-averaged moments of the dynamic arrival profile.
According to the invention the perfusion quantities are calculated from time-averaged moments of the dynamic input and arrival profiles. Such perfusion quantities represent the degree and speed of displacement of various liquids through the tissue, for example the brain of the patient to be examined, under the influence of an external effect. For example, they concern the flow of notably blood and cerebrospinal fluid through a network of arteries, veins and capillaries. The perfusion of blood and other liquids through notably the brain of the patient to be examined can be studied quantitatively on the basis of values of perfusion quantities. In order to determine the values of the perfusion quantities, for example the dynamic input and arrival profiles are measured in a time resolved manner. Such dynamic input and arrival profiles are preferably measured with a temporal resolution of 2 s or less; this means that liquid flow variations which take place at a time scale of a few seconds are reliably measured. It has been found that the perfusion quantities can be calculated from various time-averaged moments of the convolution profile. Notably deconvolution is thus avoided and surprisingly simple calculations still yield a reliable, accurate result for the perfusion quantities. The invention is based on the recognition of the fact that the perfusion quantities are simply related to time-averaged moments of the arrival profile and that the convolution relation between the arrival profile and the input profile is equivalent to an algebraic relation between the Laplace transforms of the input and arrival profiles. This means notably that a simple algebraic relation exists between the time averages of the input and arrival profiles. Moreover, a causal relation exists between the arrival profile and the input profile; this is because contrast agent will appear in the veins of the patient only after contrast agent has been introduced into an artery. As a result of this causal relation, comparatively simple algebraic relations exist between the Nth-order time-averaged moments of the arrival profile on the one side and the kth-order time-averaged moments of the input profile and the kth order time-averaged moments of the convolution profile on the other side, k and N being natural numbers and k being smaller than N. Thus, the perfusion quantities can be calculated by means of simple, notably algebraic operations. It has been found notably that the calculation of the perfusion quantities requires only rational functions, being polynomials or fractions whose numerator and denominator themselves are polynomials of the time-averaged moments of the input, arrival and convolution profiles. It is very simple to calculate such rational functions quickly and accurately. Furthermore, in respect of the input and arrival profiles the calculation of the perfusion quantities requires only the time-averaged moments of the input and arrival profiles; notably an accurate time resolved determination of the variations in time of the input and arrival profiles will not be required. According to the invention it is in particular possible to dispense with a complex and time-consuming exact determination of the arterial input, an accurate result nevertheless being obtained for the perfusion quantities.
These and other aspects of the invention will be further elucidated on the basis of the following embodiments which are defined in the dependent Claims.
A first, particularly attractive application of the invention is the calculation of the mean transit time of the contrast agent, for example through a part of the network of blood vessels in the brain. According to the invention the average transit time is calculated as the difference between the time of input and the arrival time. The times of input and arrival are simply the same as the first time-averaged moments of the dynamic input profile and arrival profile, respectively. Using the value of the average transit time of the contrast agent through the network of blood vessels, it can be quantitatively checked whether or not the blood circulation in the relevant organ, such as the brain, is in order.
A second particularly advantageous application of the invention is the calculation of the transit fraction from the input and arrival volumes. According to the invention the transit fraction is calculated as the quotient of the arrival and input volumes. The arrival and input volumes are simply calculated as the time-averaged (zero-order moments) of the respective output and input profiles. The transit fraction is a quantitative indication as to which part of the supplied liquid, such as blood containing the contrast agent, has reached the part of the body of the patient to be examined. The transit fraction, for example, accurately and quantitatively indicates whether blood circulates in the relevant organ. Furthermore, the so-called xe2x80x9cflowxe2x80x9d can be readily calculated as the quotient of the transit fraction and the average transit time. The flow represents the quantity of liquid, such as blood with the contrast agent, which flows through the relevant organ per unit of tissue of the organ to be examined and per unit of time. For example, the flow also accurately and quantitatively indicates whether the blood circulation through the relevant organ is adequate. To the physician the flow constitutes a physical quantity that is valuable in determining whether or not the relevant organ functions correctly.
It is very well possible to measure the input and arrival profiles, that is to say the time-dependent concentrations of contrast agent in the artery and the veins of the patient, respectively, by means of X-ray computed tomography, by means of a magnetic resonance imaging method or on the basis of X-ray shadow images.
The invention also relates to a magnetic resonance imaging system which includes a data processor which is arranged to carry out the method according to the invention. The data processor is, for example a computer which is programmed so as to calculate the values of the perfusion quantities. The data processor may also include a special purpose processor which is provided with electronic circuits or integrated circuits especially designed for the calculation of the values of the perfusion quantities. A magnetic resonance imaging system according to the invention acquires magnetic resonance (MR) signals from at least a part of the body of the patient to be examined. Such MR signals are generated by arranging the patient in a magnetic field and by exciting spins, notably of the protons (hydrogen nuclei) in the body of the patient by means of an RF excitation pulse; the MR signals are emitted upon relaxation of the excited spins. In order to determine the perfusion quantities, the contrast agent is administered to the patient. First magnetic resonance signals are acquired from a part of the patient which contains the arteries wherein the contrast agent is pumped via the heart. The input profile is derived from these magnetic resonance signals. When the blood with the contrast agent reaches the part of the patient to be examined, for example the organ to be examined, for example (a part of) the brain, magnetic resonance signals are acquired again from this part to be examined. The arrival profile is derived from the magnetic resonance signals from the part of the patient to be examined. The data processor in the magnetic resonance imaging system according to the invention quickly and accurately calculates the values of the perfusion quantities from the MR signals. Consequently, the magnetic resonance imaging system according to the invention is particularly suitable for a quantitative study of perfusion of liquids through the tissue, for example the brain, of the patient.